The measurement of the velocity of the body with respect to our frame necessitates the addition of a clock. We will assume that this clock beats out congruent intervals of time, where congruence is, of course, defined by the requirements of practical congruence for time, that is, by the beatings of some isolated periodic mechanism.
Now, it is perfectly apparent that unless referred to some particular frame, the path followed by a body and its motion along this path can have no determinate meaning. Thus, if in a uniformly moving train we throw a ball into the air and catch it again in our hands, then, as referred to the train, the ball will have followed an up-and-down motion along the same vertical; on the other hand, as referred to the embankment, the ball will have described a parabola. Again, if a stone is allowed to fall from a great height, its motion with respect to the earth will be accelerated, whereas, with respect to a frame of reference falling together with the stone, it will have remained at rest.
It would thus appear that we might credit to the path followed by a body any arbitrary shape we wished; all we should have to do would be to select some appropriate frame of reference. The same indeterminateness would, of course, apply to the motion of the body along its path. Thus the shape of a path in space (rectilinear or curved) or the species of a motion (uniform or accelerated) could have no absolute significance. From the standpoint of mathematical space these would all be essentially relative to the reference frame we had selected.
But it is obvious that this attitude into which our understanding of mathematical space has forced us presupposes that all the frames of reference we might select were on exactly the same footing, that is, were indistinguishable from one another. Galileo and Newton noticed that such was not the case. In certain frames we were pulled hither and thither by strange forces unsymmetrically distributed, and objects placed on the floor did not remain where they had been put. Space around us appeared to be in a chaotic condition, as, for example, on a rotating disk or in a train rounding a curve. In other frames no such forces were apparent (excluding the force of gravitation), and space appeared stagnant and quiet, the same everywhere. These latter frames are called Galilean or Inertial.
Now, a Galilean frame did not manifest itself as unique. So far as the most delicate mechanical experiments could detect, there appeared to exist an indefinite number of such frames, all moving in respect to one another with various constant speeds along straight lines, without suffering any relative rotation during their motion. Mechanical experiments conducted in any one of these Galilean frames showed that free bodies followed straight lines with constant speeds (Newton’s law of inertia), and when referred to these frames, mechanical phenomena, including the planetary motions, were susceptible of being formulated by very simple laws. As viewed from Galilean frames, the other frames, the non-Galilean ones (those filled with strange forces), were found to be moving with accelerated or rotationary motions.
Mechanical experiments conducted in the non-Galilean frames proved that the disposition of the strange unsymmetrical so-called inertial forces bore a close relationship with the apparent motions of these frames as viewed from the Galilean ones. It was possible, therefore, for an experimenter situated in one of these strange frames to anticipate, without looking outside, exactly how his frame would appear to be moving when viewed by a Galilean observer (i.e., an observer in a Galilean frame), at least so far as acceleration and rotation were concerned. Thus, in a non-Galilean frame moving with respect to a Galilean frame along a straight line with some definite acceleration, the forces of inertia would be disposed in a parallel way, pulling in the direction opposite to that of the relative acceleration. In a non-Galilean frame rotating with respect to a Galilean observer, the disposition of the inertial forces would be much more complicated. There would be the centrifugal force pulling outwards and the so-called Coriolis forces pulling sideways.
As viewed from all these non-Galilean frames, the laws of mechanics would appear hopelessly complex. Free bodies would no longer appear to follow straight lines with constant speeds, but would describe capricious curves with varying velocities. Newton’s Law of Gravitation would no longer account for the planetary motions as observed from these frames, and a study of mechanics would present tremendous difficulties. In short, all the simple laws observed from Galilean frames would have to be compounded with the effects of these strange inertial forces when we wished to formulate laws controlling phenomena as viewed from the non-Galilean frames.
No surprise need be felt, therefore, when it is said that classical science selected the Galilean frames as standard frames. All the laws of mechanics and of physics were referred to them, and this, indeed, was the essence of Copernicus’ discovery. Incidentally, we see that we are in possession of another definition of a Galilean frame; for we may say that a Galilean frame is one with respect to which the laws of classical mechanics (law of inertia, Newton’s law, etc.) hold true.
We may also mention that the very possibility of there being definite laws of mechanics, such as the law of inertia, is contingent upon the existence of certain fundamental differences between the various frames. If all frames were identical, as the complete relativity of motion would demand, the law of inertia would express nothing at all. In every case we might make the motion of a body appear anything we wished by selecting our frame suitably; and if we had no means of distinguishing one frame from another the straight path of a body through empty space would convey no physical meaning.
We may now give still another definition of a Galilean frame. It follows from the law of inertia, according to which free bodies, when viewed from Galilean frames, will describe straight lines with constant speeds, that the stars, being presumably free bodies, will obey this law and, owing to their remoteness, appear to suffer very slight displacements even in the course of a century This fact led to a more easily obtainable definition of a Galilean or inertial frame, namely, a frame with respect to which the stars would appear fixed. This definition is not meant to supplant the one given previously of a frame in which no forces of inertia are experienced, for theoretically both definitions are equivalent. Its sole advantage is that it leads to a more accurate empirical determination of a Galilean frame and shows us immediately that our earth does not constitute such a frame, since the stars appear to circle round the Pole star. As viewed from a true Galilean frame, our earth would therefore be rotating round the polar axis every twenty-four hours.